Mathematical analysis

Short description: Branch of mathematics

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

Mathematical analysis is the branch of mathematics that studies functions, spaces, and operators through quantitative methods of approximation and convergence. It grew out of calculus, especially the use of derivatives and integrals to study variable quantities, and in the 19th century its foundations were reformulated with greater rigor. Basic objects of study in mathematical analysis include the real numbers, functions, sequences, series, and limits. Analysis has remained closely connected with applications in the sciences, where it is used to study equations, approximate one object by another, and estimate the accuracy of such approximations.

Modern analysis studies these questions in many settings, including Euclidean spaces, metric spaces, topological spaces, measure spaces, and function spaces. Its major areas include complex analysis, functional analysis, measure theory, harmonic analysis, and the theory of ordinary and partial differential equations.

## Contents

* 1 History
* 1.1 Ancient
* 1.2 Medieval
* 1.3 Renaissance
* 1.4 Modern
* 1.4.1 Foundations
* 1.4.2 Modernization
* 2 Important concepts
* 2.1 Real numbers
* 2.2 Approximation and convergence
* 2.3 Continuity
* 2.4 Metric spaces
* 2.5 Complex variables
* 2.6 Measures, averaging, and probability
* 2.7 Decomposing functions into simpler pieces
* 2.8 Dynamics and evolution
* 2.9 Operators and spectral theory
* 3 Main branches
* 3.1 Calculus
* 3.2 Real analysis
* 3.3 Complex analysis
* 3.4 Functional analysis
* 3.5 Fourier analysis
* 3.6 Harmonic analysis
* 3.7 Differential equations
* 3.8 Measure theory
* 3.9 Numerical analysis
* 3.10 Geometric analysis and global analysis
* 3.11 Convex analysis
* 3.12 Calculus of variations
* 3.13 Probability theory and stochastic analysis
* 4 Other topics
* 5 Applications
* 5.1 Physical sciences
* 5.2 Signal processing
* 5.3 Other areas of mathematics
* 6 Notable textbooks
* 7 See also
* 8 References
* 9 Further reading
* 10 External links

## History

Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

### Ancient

Mathematical analysis formally developed in the 17th century during the Scientific Revolution,[1] but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy.[2] (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.[3] The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.[4] In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle.[5] From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE.[6] Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE.[7]

### Medieval

During the medieval period, mathematicians in several traditions developed methods that anticipated later topics in analysis, including quadrature, infinite series, infinitesimal reasoning, approximation, and the mathematical study of motion. In the Islamic world, Ibn al-Haytham worked on sums of powers and area problems, and Ibrahim ibn Sinan generalized Archimedean methods in the quadrature of the parabola.[8][9] In medieval Europe, the Oxford Calculators studied uniformly accelerated motion, and Nicole Oresme gave a graphical proof of the mean speed theorem, representing displacement by the area under a velocity-time graph. Oresme also used infinite series and proved the divergence of the harmonic series.[10] In India, Bhāskara II used infinitesimal reasoning and stated results related to what is now called Rolle's theorem, and the Kerala school of astronomy and mathematics, especially Madhava, developed infinite series for trigonometric functions.[11] These developments were not mathematical analysis in the modern sense, but were important precursors to calculus and analysis.

Questions of the nature of the continuum were important to medieval European philosophers. In particular, whether the continuum could be infinitely divided, whether it consisted of points. The works of Aristotle, which only became more widely available in Europe in the earth 13th century, held that the continuum was not made of points: if it were, the points would have to be side-by-side which would be incompatible with the inseparability of the continuum. In the 14th century, Thomas Bradwardine described the continuum as being made of an infinite collection of infinitesimals in Tractatus de continuo, but not of points. William of Occam held, contrarily, that the continuum is made of actual points. Aristotle had also written on the nature of infinity, classifying two different kinds of infinity: potential infinity and actual infinity. For medieval European philosophers, the nature of infinity posed problems for the ideas of infinite divisibility and infinitesimals, because if a subdivision of a continuum was infinitesimal, its ratio to the whole continuum would need to be infinite, and various paradoxes would result. Richard Swineshead in Liber calculationum wrote that such ratios should simply be left undefined, and that the arguments about infinity do not proceed as arguments about finite quantities. These are some of the earliest seeds in the European tradition in which Leibniz's views on the continuum would emerge some three centuries years later, with Leibniz explicitly crediting Swineshead.[12]

### Renaissance

In the early seventeenth century, methods that anticipated the calculus became increasingly systematic. Johannes Kepler used infinitesimal and summation arguments in problems of area and volume, Galileo Galilei connected mathematics with the study of motion, Bonaventura Cavalieri developed the method of indivisibles, and Evangelista Torricelli extended such methods in geometry and mechanics. These works did not yet constitute mathematical analysis in the modern sense, but they helped shift the subject from classical geometric constructions toward general methods for treating continuous quantities, tangents, areas, volumes, and motion. The subsequent development of differential and integral calculus by Newton and Leibniz became the starting point for much of later analysis.[13]

### Modern

#### Foundations

The modern foundations of mathematical analysis were established in 17th century Europe.[1] This began when Fermat and Descartes developed analytic geometry, which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves.[14] Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered to be the establishment of mathematical analysis. It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

#### Modernization

In the 18th century, Euler introduced the notion of a mathematical function.[15] Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816,[16] but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration, and made significant advances in complex analysis.

Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.

Also, various pathological objects, (such as nowhere continuous functions, continuous but nowhere differentiable functions, and space-filling curves), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration, which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

## Important concepts

### Real numbers

The real numbers provide the standard setting for much of classical analysis. Their completeness, often expressed by the least-upper-bound property, underlies basic results about limits, continuity, differentiation, and integration.

### Approximation and convergence

Approximation plays a fundamental role in many areas of mathematics. An example is the limit of a sequence of real numbers. A sequence of real numbers is a family a1,a2… of real numbers, each indexed by a natural number. A sequence is said to converge to a limit L if almost all members of the sequence are arbitrarily close to L. More precisely, this means that for any error tolerance ϵ, all of the members of the sequence are within ϵ of L except possibly for finitely many members of the sequence. Formally, for any error ϵ>0 there is an integer N such that |an−L|<ϵ whenever n>N.

This example shows the use of approximation: the elements an approximate the number L, and the error tolerance is ϵ. However, convergence alone does not provide information on how good the approximation is, and many results in analysis concern the quality of approximation.

Another example comes from differential calculus. A real function f is differentiable at a point a if there is a linear function L(x)=f(a)+f′(a)(x−a) that approximates the function well near the point x=a. But "well" here only means that the approximation error |f(x)−L(x)|=o(x−a) where o(x−a) is a function that tends to zero faster than x−a as x->a. Better approximations provide more uniform and quantitative estimates on the size of the error term. Taylor's theorem, for example, states that for a twice continuously-differentiable function on a closed interval containing a |f(x)−L(x)|≤M(x−a)2 where M can be estimated explicitly using the second derivative. This allows the error in the linear approximation to determined much more precisely than differentiability at the point.

The inequality |f(x)−L(x)|≤M(x−a)2 is an example of what is called an estimate in analysis. Estimates are inqualities that are used to quantify the error in an approximation, as well as more generally to express that a certain operation is controlled (bounded) by some other operation.

### Continuity

Continuity also plays a key role in analysis. In elementary analysis, the idea of a continuous function is introduced using an epsilon-delta definition. Roughly, a function is continuous at a point if sufficiently small changes in the input produce small changes in the output. This rules out "jumps" in the graph of the function, or other kinds of pathological oscillatory behavior.

Continuity is important in analysis because it allows local control of a function to imply global conclusions when combined with additional hypotheses such as connectedness or compactness. For example, continuous real-valued functions on intervals have the intermediate value property, and continuous real-valued functions on compact sets attain maximum and minimum values. Continuous functions on compact metric spaces are also uniformly continuous, meaning that the function oscillates on a comparable scale throughout the domain. These results are basic tools in calculus, optimization, differential equations, and approximation theory.

Continuous functions generalize readily to metric spaces and other topological spaces. Spaces of continuous functions are among the most studied in functional analysis, where they provide useful structural probes of a space. In differential equations, continuity is also a threshold regularity condition: in more advanced analysis, regularity theorems often show that objects first defined only weakly, almost everywhere, or distributionally have continuous representatives under additional hypotheses, and can thus be treated as honest functions rather than more general objects.

### Metric spaces

Main article: Metric space

A metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers.

Much of functional analysis is concerned with spaces of functions, which can be given the structure of a metric space, such as Banach spaces and Hilbert spaces. In many of these examples, the metric comes from a norm. For example, the space of continuous real-valued functions C([0,1]) on the unit interval is a Banach space under the supremum norm. The spaces of primary importance in measure theory and harmonic analysis are the Lp spaces, which are metric spaces whose metrics again come from a norm, are complete; i.e., they are Banach spaces.

Metric spaces are extremely convenient in analysis because many of the strong approximation results that hold in Euclidean space carry over to metric spaces with minimal changes. For example, compactness in metric spaces is equivalent to sequential compactness, so limiting arguments on metric spaces can be comparatively straightforward in metric spaces as opposed to in more general spaces in analysis. Compact metric spaces and complete metric spaces are especially important in analysis, where many arguments require the existence of limits.

### Complex variables

Main article: Complex analysis

Complex numbers provide another important tool for many analytic equations. A complex-valued function of a complex variable is holomorphic if its derivative exists as a complex limit at each point of its domain. Holomorphic functions are much more rigid than differentiable functions of a real variable: they are analytic functions, represented locally by convergent power series.

An important tool in complex variables is contour integration, in which functions are integrated along curves in the complex plane. The Cauchy integral theorem, Cauchy integral formula, and residue theorem relate the values of a holomorphic or meromorphic function to its behavior on curves and near singularities. These results allow one to shift contours when integrating a holomorphic function, provided the deformation of the contour does not cross a singularity. This is useful in evaluating many real integrals, and in the study of functions through their singularities.

In operator theory and spectral theory, the resolvent of an operator encodes information about its spectrum and often allows functions of operators to be defined by complex integration. Complex variables thus appear in connection with differential and integral equations, eigenvalue problems, and the theory of linear operators.

### Measures, averaging, and probability

Measure theory gives a systematic way of assigning sizes to subsets of a space, in a way that generalizes length, area, and volume in Euclidean space. Abstractly, measure theory begins by specifying a class of sets which are measurable, that is, sets that have a measure. The measurable sets form a sigma algebra, meaning that one can take countable unions and intersections, as well as set complements. Measures are then defined in such a way as to be compatible with the set operations on the measurable sets.

In measure theory, pointwise convergence of functions can be replaced with the notion of convergence almost everywhere, that is, convergence at every point except a set whose measure is zero. Convergence almost everywhere is much more convenient in measure theory: often various types of mean convergence cannot control a precise set on which pointwise convergence fails, but can ensure that the bad set has measure zero. Functions are often identified if they agree almost everywhere, that is, off a set of measure zero. Thus measure zero sets are often in practice simply ignored.

The Lebesgue integral extends the Riemann integral, and is better adapted to the limiting processes of analysis. The idea of the Lebesgue integral, for a non-negative function f, is that it possible to form a rearrangement of its values to form a decreasing function, using the measure of its superlevel sets to define the rearrangement. Integration of decreasing functions have better limit properties under integration, and that good behavior can be transferred to the Lebesgue integral. One has theorems like monotone convergence, Fatou's lemma, and the dominated convergence theorem that simplify many limit arguments.

Measure theory also supplies the underlying mathematics of probability theory. A probability space is a measure space whose total measure is one, and expected values are integrals with respect to this probability measure.

Many function spaces in analysis are defined using measures. The Lp spaces consist of functions whose powers are integrable, with functions identified when they agree almost everywhere. These spaces are important throughout analysis, such as in harmonic analysis and partial differential equations.

### Decomposing functions into simpler pieces

Another theme in analysis is that of decomposing functions into simpler pieces, determined by symmetry, scale, or oscillation. The prototypical example is that of Fourier series, which decomposes a periodic function into basic sinusoids. The Fourier series takes the form of a trigonometric series ∑n=−∞∞cneinθ where cn are complex numbers and θ is the independent variable.

The Fourier series is one instance of an eigenfunction expansion, with the exponentials einθ being the eigenfunctions of the rotation group acting on the circle. It has the property of being an orthogonal expansion: any two of the eigenfunctions are orthogonal in the Hilbert space of square integrable functions on the circle. Eigenfunction expansions appear in many areas in mathematical analysis, and particularly in its applications to the sciences where symmetry is often important. An example of a different sort of eigenfunction expansion is the decomposition of a function on the sphere into spherical harmonics. Again, this is an orthogonal expansion, and orthogonality of the expansion leads to a tractable isolation of each space.

Expansions can be used to study convergence and approximation, smoothness and oscillation, decay, and solutions of differential equations. Sobolev spaces, for example, relate the smoothness of functions to the decay of their Fourier coefficients. These representations can exhibit structural information that may be difficult to see from the original form of a function.

Fourier analysis is the study of such decompositions, chiefly focused on the Fourier transform and some of its generalizations.

### Dynamics and evolution

Main articles: Dynamical system, Differential equation, Ergodic theory, and C0 semigroup

Many problems in analysis concern how quantities change over time or under repeated application of a rule. This leads to ordinary differential equations and partial differential equations, where one studies functions whose derivatives satisfy prescribed relations. For repeated applications of a rule, one studies iterates of a function or transformation, giving rise to discrete dynamical systems.

Analytic questions in dynamics include existence and uniqueness of solutions, stability, approximation of trajectories, long-time behavior, and dependence on initial conditions. For example, a differential equation may determine a flow on a space, while iteration of a map produces an orbit. Analytic tools are used to determine whether such orbits converge, remain bounded, become periodic, or exhibit more complicated behavior.

Dynamics is an important application of analytic methods, and leads to ideas and techniques within analysis itself. In ergodic theory, the key objects are transformations that preserve a measure, and one asks how time averages along orbits relate to space averages over the whole system. The basic theory concerns whether time averages of functions become equal to their spatial averages under the orbit of a system, and how rapidly that approximation takes place. In functional analysis, evolution problems are often studied using one-parameter families of operators, such as operator semigroups, which generalize the exponential function from numbers or matrices to infinite-dimensional spaces.

### Operators and spectral theory

Many areas of analysis study operators, like differential operators, integral operators, or linear transformations on a function space or other topological vector space. Operators can encode information such as the evolution of a system, a differential or integral equation, or a quantum state or observable. The spectral theory of operators allows operators to be broken into pieces and represented, generalizing aspects of the eigenvalue decomposition from linear algebra to infinite dimensions. As in linear algebra, it is often possible to understand an operator more deeply through its spectral decomposition.

## Main branches

### Calculus

Main article: Calculus

Calculus is a branch of analysis that deals primarily with calculation and applications.[17][18] The two branches of calculus are: integral calculus, which deals with the study of averages, accumulation, and area; and differential calculus, which studies rates of change, linear approximation, and the derivative. Examples of applications of calculus include how to work with approximations such as Taylor approximations and Taylor series, integration in elementary terms, numerical approximations to integration, and basic optimization.

Vector analysis, also called vector calculus, is part of calculus that deals with vector-valued functions.[19]

### Real analysis

Main article: Real analysis

Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.[20][21] In particular, it deals with the properties of real functions and sequences, including convergence and limits of sequences of real numbers, providing rigorous foundations for calculus, and continuity, smoothness and related properties of real-valued functions.

### Complex analysis

Main article: Complex analysis

Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers.[22][23][24] It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.

Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.

### Functional analysis

Main article: Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.[25][26][27][28] The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

### Fourier analysis

Main article: Fourier analysis

Fourier analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves.[29][30] This includes the study of the notions of Fourier series and Fourier transforms, and of their generalizations. Fourier analysis also includes the study of linear differential operators with constant coefficients and their generalizations to pseudodifferential operators.

Microlocal analysis is a subfield of Fourier analysis concerned with how to localize the singularities of functions, and the how they propagate under differential and pseudodifferetial operators.

### Harmonic analysis

Main article: Harmonic analysis

Harmonic analysis is a branch of mathematics having origins in the study of harmonic functions, and especially their boundary values. Harmonic function theory leads naturally to the study of function spaces like Hardy spaces, and in particular criteria for membership in such spaces and estimates of opeerators. It includes many of the methods of Fourier analysis, but also other decomposition methods such as Calderón–Zygmund decompositions which break functions into parts that can be handled by Fourier methods and parts that can be handled by local methods.[31]

Abstract harmonic analysis is a related tradition, which generalizes Fourier methods to groups other than those to which classical Fourier methods apply.

### Differential equations

Main article: Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.[32][33][34][35] Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

### Measure theory

Main article: Measure (mathematics)

A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.[36][37] In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space ℝn. For instance, the Lebesgue measure of the interval [0,1] in the real numbers is its length in the everyday sense of the word – specifically, 1.

Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X. It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that the empty set, countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.

### Numerical analysis

Main article: Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).[38]

Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

### Geometric analysis and global analysis

Main articles: Geometric analysis and Global analysis

Geometric analysis is the branch of analysis concerned with the study of manifolds, often Riemannian manifolds. Global questions of Riemannian geometry are often studied.[39] One example is the spectral geometry of the Laplace–Beltrami operator, which generalizes the problem of hearing the shape of a drum, for instance. Other global questions in geometric analysis include partial differential equations on Riemannian manfiolds, such as the Yamabe equation and the Ricci flow. The latter led to the proof of the Poincaré conjecture.

### Convex analysis

Main article: Convex analysis

Convex analysis is the branch of analysis concerned with convex functions, convex sets, and applications to optimization and linear programming. Convex functions generally have more powerful methods available for ensuring the existence and uniqueness of minima, particularly when combined with lower semicontinuity. There are many dualities in convex optimization, often expressed in terms of the convex conjugate, which allow an optimization problem to be paired with a dual problem, which gives a useful criterion for checking optimality of a putative solution or determining how much slack there is in a numerical solution.

### Calculus of variations

Main article: Calculus of variations

Calculus of variations[40] is the study of finding optimal solutions to minimization problems in infinite dimensional spaces. A basic example is to find a geodesic on a surface: the endpoints are given, and one must find a path among the infinite-dimensional space of all possible paths, that minimizes the arc length. Many problems in partial differential equations admit a more natural variational characterization, and this can lead to notions of weak solutions, which are often more suited to analytic methods and numerical work. Direct methods in the calculus of variations formulate variational problems as convex optimization problems.

Optimal transport is the sub-branch of the calculus of variations concerned with the formulation and solution of the problem of transporting stuff from one place to another subject to cost constraints. This sub-branch also shares many ideas with convex analysis, including duality theorems such as Kantorovich duality, which is a form of Legendre duality. The ideas of optimal transportation have wide and unexpected applications, such as the Wasserstein metric and its applications to machine learning.

### Probability theory and stochastic analysis

Main article: Probability theory

Probability theory is closely connected with mathematical analysis through measure theory.[41] In the modern axiomatic formulation, a probability space is a measure space of total measure one, random variables are measurable functions, and expected values are integrals with respect to a probability measure. Analytic methods are central in the study of convergence of random variables, martingales, stochastic processes, Brownian motion, stochastic differential equations, and ergodic theory.

Probability theory is an area of mathematics in its own right, but many of its modern foundations and methods belong to analysis. Stochastic analysis studies analytic questions involving random processes, including stochastic integration, stochastic differential equations, and connections with partial differential equations and functional analysis.

## Other topics

* Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.
* p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
* Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.
* Computable analysis, the study of which parts of analysis can be carried out in a computable manner.
* Set-valued analysis – applies ideas from analysis and topology to set-valued functions.
* Idempotent analysis – analysis in the context of an idempotent semiring, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A.
* Tropical analysis – analysis of the idempotent semiring called the tropical semiring (or max-plus algebra/min-plus algebra).
* Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory.
* Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences.
* Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.
* Smooth infinitesimal analysis, which is developed in a smooth topos.
* Dynamical systems, which is the study of the behavior of systems under time-evolution.

## Applications

Techniques from analysis are also found in other areas such as:

### Physical sciences

The vast majority of classical mechanics, relativity, and quantum mechanics is based on applied analysis, and differential equations in particular. Examples of important differential equations include Newton's second law, the Schrödinger equation, and the Einstein field equations.

Functional analysis is also a major factor in quantum mechanics.

### Signal processing

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.[42]

### Other areas of mathematics

Techniques from analysis are used in many areas of mathematics, including:

* Analytic number theory
* Analytic combinatorics
* Continuous probability
* Differential entropy in information theory
* Differential games
* Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
* Differentiable manifolds
* Differential topology
* Partial differential equations

## Notable textbooks

* Leonhard Euler (1748) Introductio in analysin infinitorum
* A. L. Cauchy (1821) Cours d'analyse
* Camille Jordan (1882) Cours d'analyse de l'École polytechnique
* G. H. Hardy (1908) A Course of Pure Mathematics
* E. T. Whittaker & G. N. Watson (1915) A Course of Modern Analysis
* George Pólya & Gábor Szegő (1925) Problems and Theorems in Analysis (two volumes)[43][44]
* Walter Rudin (1953) Principles of Mathematical Analysis[45]
* Elias M. Stein & Rami Shakarchi (2003, 2011) Princeton Lectures in Analysis (four volumes)

## See also

* Arithmetization of analysis
* Constructive analysis
* History of calculus
* Hypercomplex analysis
* Multiple rule-based problems
* Multivariable calculus
* Paraconsistent logic
* Smooth infinitesimal analysis
* Timeline of calculus and mathematical analysis

## References

1. ↑ 1.0 1.1 Jahnke, Hans Niels (2003). A History of Analysis. History of Mathematics. 24. American Mathematical Society. p. 7. doi:10.1090/hmath/024. ISBN 978-0821826232. https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7. Retrieved 2015-11-15.
2. ↑ "Infinite Series". Mathematics and its History (2nd ed.). Springer Science+Business Media Inc.. 2004. p. 170. ISBN 978-0387953366. "Infinite series were present in Greek mathematics, [...] There is no question that Zeno's paradox of the dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series 1⁄2 \+ 1⁄22 \+ 1⁄23 \+ 1⁄24 \+ ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + 1⁄4 \+ 1⁄42 \+ 1⁄43 \+ ... = 4⁄3. Both these examples are special cases of the result we express as summation of a geometric series"
3. ↑ History of Mathematics. Dover Publications. 1958. ISBN 978-0486204307. https://archive.org/details/historyofmathema0002smit.
4. ↑ Pinto, J. Sousa (2004). Infinitesimal Methods of Mathematical Analysis. Horwood Publishing. p. 8. ISBN 978-1898563990. https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7. Retrieved 2015-11-15.
5. ↑ Dun, Liu; Fan, Dainian; Cohen, Robert Sonné (1966). A comparison of Archimedes' and Liu Hui's studies of circles. Chinese studies in the history and philosophy of science and technology. 130. Springer. p. 279. ISBN 978-0-7923-3463-7. https://books.google.com/books?id=jaQH6_8Ju-MC. Retrieved 2015-11-15. , Chapter, p. 279
6. ↑ Singh, A. N. (1936). "On the Use of Series in Hindu Mathematics". Osiris 1: 606–628. doi:10.1086/368443. https://www.jstor.org/stable/301627.
7. ↑ K. B. Basant, Satyananda Panda (2013). "Summation of Convergent Geometric Series and the concept of approachable Sunya". Indian Journal of History of Science 48: 291–313. https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol48_2_7_KBBasant.pdf.
8. ↑ Katz, Victor J. (1995). "Ideas of Calculus in Islam and India". Mathematics Magazine 68 (3): 163–174. doi:10.2307/2691411.
9. ↑ O'Connor, John J.; Robertson, Edmund F.. "Ibrahim ibn Sinan". University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Ibrahim/.
10. ↑ Stillwell, John Colin. "Analysis: Models of motion in medieval Europe". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics/Models-of-motion-in-medieval-Europe. Retrieved 19 June 2026.
11. ↑ Seal, Brajendranath (1916), "The positive sciences of the ancient Hindus", Nature 97 (2426): 177, doi:10.1038/097177a0, Bibcode: 1916Natur..97..177.
12. ↑ Boyer, Carl B. (1959). The History of the Calculus and Its Conceptual Development. New York: Dover Publications. pp. 66–70.
13. ↑ Boyer, Carl B. (1959). The History of the Calculus and Its Conceptual Development. New York: Dover Publications. pp. 101–150.
14. ↑ Pellegrino, Dana. "Pierre de Fermat". http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html.
15. ↑ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. p. 17. https://archive.org/details/eulermasterofusa0000dunh.
16. ↑ *Cooke, Roger (1997). "Beyond the Calculus". The History of Mathematics: A Brief Course. Wiley-Interscience. p. 379. ISBN 978-0471180821. https://archive.org/details/historyofmathema0000cook/page/379. "Real analysis began its growth as an independent subject with the introduction of the modern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781–1848)"
17. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
18. ↑ Mathematics: Its Content, Methods, and Meaning. 1. Cambridge, Massachusetts: MIT Press. 1963. , Chapter 1
19. ↑ Borisenko, A. I.; Tarapov, I. E. (1979). Vector and Tensor Analysis with Applications (Dover Books on Mathematics). Dover Books on Mathematics.
20. ↑ Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw–Hill. ISBN 978-0070542358. https://archive.org/details/principlesofmath00rudi.
21. ↑ Abbott, Stephen (2001). Understanding Analysis. Undergraduate Texts in Mathematics. New York: Springer-Verlag. ISBN 978-0387950600.
22. ↑ Complex Analysis (3rd ed.). New York: McGraw-Hill. 1979. ISBN 978-0070006577. https://books.google.com/books?id=2MRuus-5GGoC.
23. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
24. ↑ Mathematics: Its Content, Methods, and Meaning. 1. Cambridge, Massachusetts: MIT Press. 1963. , Chapter 1
25. ↑ Functional Analysis. McGraw-Hill Science. 1991. ISBN 978-0070542365. https://archive.org/details/functionalanalys0000rudi.
26. ↑ A Course in Functional Analysis (2nd ed.). Springer-Verlag. 1994. ISBN 978-0387972459. https://books.google.com/books?id=ix4P1e6AkeIC. Retrieved 2016-02-11.
27. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
28. ↑ Mathematics: Its Content, Methods, and Meaning. 1. Cambridge, Massachusetts: MIT Press. 1963. , Chapter 1
29. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
30. ↑ Stein, Elias M.; Shakarchi, Rami (2005). Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis. 3. Princeton, New Jersey: Princeton University Press. p. vii. ISBN 978-0-691-11386-9.
31. ↑ Stein, Elias M.; Shakarchi, Rami (2011). Functional Analysis: Introduction to Further Topics in Analysis. Princeton Lectures in Analysis. 4. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-11387-6.
32. ↑ Ince, Edward L. (1956). Ordinary Differential Equations. Dover Publications. ISBN 978-0486603490. https://books.google.com/books?id=mbyqAAAAQBAJ.
33. ↑ Witold Hurewicz, Lectures on Ordinary Differential Equations, Dover Publications, ISBN 0486495108
34. ↑ Partial Differential Equations. Providence: American Mathematical Society. 1998. ISBN 978-0821807729.
35. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
36. ↑ Tao, Terence (2011). An Introduction to Measure Theory. Graduate Studies in Mathematics. 126. American Mathematical Society. doi:10.1090/gsm/126. ISBN 978-0821869192. https://books.google.com/books?id=HoGDAwAAQBAJ. Retrieved 2018-10-26.
37. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
38. ↑ Hildebrand, Francis B. (1974). Introduction to Numerical Analysis (2nd ed.). McGraw-Hill. ISBN 978-0070287617.
39. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
40. ↑ Stillwell, John Colin. "Analysis". Encyclopaedia Britannica. Encyclopaedia Britannica, Inc.. https://www.britannica.com/science/analysis-mathematics. Retrieved 18 June 2026.
41. ↑ Stein, Elias M.; Shakarchi, Rami (2011). Functional Analysis: Introduction to Further Topics in Analysis. Princeton Lectures in Analysis. 4. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-11387-6.
42. ↑ Rabiner, L. R.; Gold, B. (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 978-0139141010. https://archive.org/details/theoryapplicatio00rabi.
43. ↑ Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions. ASIN 3540636404.
44. ↑ Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry. ASIN 3540636862.
45. ↑ Principles of Mathematical Analysis. ASIN 0070856133.

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## Further reading

* Mathematics: Its Content, Methods, and Meaning. 1–3 (2nd ed.). Cambridge, Massachusetts: The M.I.T. Press / American Mathematical Society. March 1969.
* Mathematical Analysis (2nd ed.). Addison–Wesley. 1974. ISBN 978-0201002881.
* The foundations of analysis: a straightforward introduction. Cambridge University Press. 1981. https://archive.org/details/foundationsofana0000binm.
* Foundations of mathematical analysis. New York: M. Dekker. 1981.
* Hazewinkel, Michiel, ed (2002). "Mathematical analysis". Encyclopaedia of Mathematics. Springer-Verlag. ISBN 978-1402006098. https://encyclopediaofmath.org/wiki/Mathematical_analysis.
* (in it) Analisi Matematica Due. Liguori Editore (it). 1996. ISBN 978-8820726751.
* (in fr) Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques. EDP Sciences. 2004. ISBN 978-2868836816.
* Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. 1976. ISBN 978-0070542358.
* Real and Complex Analysis (3rd ed.). New York: McGraw-Hill. 1987. ISBN 978-0070542341.
* A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th ed.). Cambridge: at the University Press. 1927-01-02. ISBN 0521067944. (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992)
* "Real Analysis – Course Notes". http://www.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf.

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